Allegory (category Theory)

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## Definition

## Regular categories and allegories

### Allegories of relations in regular categories

### Maps in allegories, and tabulations

### Unital allegories and regular categories of maps

### More sophisticated kinds of allegory

## References

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

Allegory Category Theory

In the mathematical field of category theory, an **allegory** is a category that has some of the structure of the category **Rel** of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in this sense the theory of allegories is a generalization of relation algebra to relations between different sorts. Allegories are also useful in defining and investigating certain constructions in category theory, such as exact completions.

In this article we adopt the convention that morphisms compose from right to left, so *RS* means "first do S, then do R".

An allegory is a category in which

- every morphism is associated with an
**anti-involution**, i.e. a morphism with and and - every pair of morphisms with common domain/codomain is associated with an
**intersection**, i.e. a morphism

all such that

- intersections are idempotent: commutative: and associative:
- anti-involution distributes over intersection:
- composition is semi-distributive over intersection: and and
- the modularity law is satisfied:

Here, we are abbreviating using the order defined by the intersection: means

A first example of an allegory is the category of sets and relations. The objects of this allegory are sets, and a morphism is a binary relation between X and Y. Composition of morphisms is composition of relations; intersection of morphisms is intersection of relations.

In a category C, a **relation** between objects X and Y is a span of morphisms that is jointly monic. Two such spans and are considered equivalent when there is an isomorphism between S and T that make everything commute; strictly speaking, relations are only defined up to equivalence (one may formalise this either by using equivalence classes or by using bicategories). If the category C has products, a relation between X and Y is the same thing as a monomorphism into *X* × *Y* (or an equivalence class of such). In the presence of pullbacks and a proper factorization system, one can define the composition of relations. The composition is found by first pulling back the cospan and then taking the jointly-monic image of the resulting span

Composition of relations will be associative if the factorization system is appropriately stable. In this case, one can consider a category Rel(*C*), with the same objects as C, but where morphisms are relations between the objects. The identity relations are the diagonals

A regular category (a category with finite limits and images in which covers are stable under pullback) has a stable regular epi/mono factorization system. The category of relations for a regular category is always an allegory. Anti-involution is defined by turning the source/target of the relation around, and intersections are intersections of subobjects, computed by pullback.

A morphism R in an allegory A is called a **map** if it is entire and deterministic Another way of saying this is that a map is a morphism that has a right adjoint in A when *A* is considered, using the local order structure, as a 2-category. Maps in an allegory are closed under identity and composition. Thus, there is a subcategory Map(*A*) of A with the same objects but only the maps as morphisms. For a regular category C, there is an isomorphism of categories In particular, a morphism in Map(Rel(**Set**)) is just an ordinary set function.

In an allegory, a morphism is **tabulated** by a pair of maps and if and An allegory is called **tabular** if every morphism has a tabulation. For a regular category C, the allegory Rel(*C*) is always tabular. On the other hand, for any tabular allegory A, the category Map(*A*) of maps is a locally regular category: it has pullbacks, equalizers, and images that are stable under pullback. This is enough to study relations in Map(*A*), and in this setting,

A **unit** in an allegory is an object U for which the identity is the largest morphism and such that from every other object, there is an entire relation to U. An allegory with a unit is called **unital**. Given a tabular allegory A, the category Map(*A*) is a regular category (it has a terminal object) if and only if A is unital.

Additional properties of allegories can be axiomatized. **Distributive allegories** have a union-like operation that is suitably well-behaved, and **division allegories** have a generalization of the division operation of relation algebra. **Power allegories** are distributive division allegories with additional powerset-like structure. The connection between allegories and regular categories can be developed into a connection between power allegories and toposes.

- Peter Freyd, Andre Scedrov (1990).
*Categories, Allegories*. Mathematical Library Vol 39. North-Holland. ISBN 978-0-444-70368-2.

- Peter Johnstone (2003).
*Sketches of an Elephant: A Topos Theory Compendium*. Oxford Science Publications. OUP. ISBN 0-19-852496-X.

This article uses material from the Wikipedia page available here. It is released under the Creative Commons Attribution-Share-Alike License 3.0.

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